Difference between revisions of "MAT1224"
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* [[Differentiation Rules]] <!-- 1214-3.3 --> | * [[Differentiation Rules]] <!-- 1214-3.3 --> | ||
− | * [[The Chain Rule]] <!-- 1214-3.6 --> | + | * [[Chain Rule|The Chain Rule]] <!-- 1214-3.6 --> |
* [[Antiderivatives]] <!-- 1214-4.10 --> | * [[Antiderivatives]] <!-- 1214-4.10 --> | ||
* [[The Definite Integral]] <!-- 1214-5.2 --> | * [[The Definite Integral]] <!-- 1214-5.2 --> | ||
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[[Integration by Substitution]] | [[Integration by Substitution]] | ||
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* [[Differentiation Rules]] <!-- 1214-3.3 --> | * [[Differentiation Rules]] <!-- 1214-3.3 --> | ||
− | * [[Differentials]] <!-- 1214-4.2 --> | + | * [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 --> |
* [[Antiderivatives]] <!-- 1214-4.10 --> | * [[Antiderivatives]] <!-- 1214-4.10 --> | ||
* [[The Definite Integral]] <!-- 1214-5.2 --> | * [[The Definite Integral]] <!-- 1214-5.2 --> | ||
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
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+ | * Recognize when to use integration by substitution. | ||
* Use substitution to evaluate indefinite integrals. | * Use substitution to evaluate indefinite integrals. | ||
* Use substitution to evaluate definite integrals. | * Use substitution to evaluate definite integrals. | ||
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[[Area between Curves]] | [[Area between Curves]] | ||
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− | * [[Toolkit Functions| | + | * [[Toolkit Functions|Graphing Elementary Functions]] <!-- 1073-Mod 1.2 --> |
− | * [[ | + | * [[The Definite Integral]] <!-- 1214-5.2 --> |
+ | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
+ | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
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− | * Find the area of | + | * Determine the area of a region between two curves by integrating with respect to the independent variable. |
+ | * Find the area of a compound region. | ||
+ | * Determine the area of a region between two curves by integrating with respect to the dependent variable. | ||
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− | * [[Toolkit Functions| | + | * [[Areas of basic shapes]] <!-- Grades 6-12 --> |
− | * | + | * [[Volume of a cylinder]] <!-- Grades 6-12 --> |
− | * [[ | + | * [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 --> |
+ | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
+ | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
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− | * | + | * Determine the volume of a solid by integrating a cross-section (the slicing method). |
− | + | * Find the volume of a solid of revolution using the disk method. | |
− | + | * Find the volume of a solid of revolution with a cavity using the washer method. | |
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− | * [[Toolkit Functions| | + | * [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 --> |
− | * [[The | + | * [[Determining Volumes by Slicing]] <!-- 1224-2.2 --> |
+ | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
+ | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
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− | * | + | * Calculate the volume of a solid of revolution by using the method of cylindrical shells. |
− | + | * Compare the different methods for calculating a volume of revolution. | |
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[[Arc Length and Surface Area]] | [[Arc Length and Surface Area]] | ||
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− | * [[ | + | * [[Differentiation Rules]] <!-- 1214-3.3 --> |
− | * [[The | + | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> |
− | + | * [[Integration by Substitution]] <!-- 1224-1.5 --> | |
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+ | * Determine the length of a plane curve between two points. | ||
+ | * Find the surface area of a solid of revolution. | ||
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− | * [[ | + | * [[Areas of basic shapes]] <!-- Grades 6-12 --> |
− | * | + | * [[Volume of a cylinder]] <!-- Grades 6-12 --> |
+ | * [[Basic Physics (Mass, Force, Work, Newton's Second Law, Hooke's Law)]] <!-- Grades 6-12 --> | ||
+ | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
+ | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
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− | * | + | * Determine the mass of a one-dimensional object from its linear density function. |
− | * | + | * Determine the mass of a two-dimensional circular object from its radial density function. |
− | * | + | * Calculate the work done by a variable force acting along a line. |
− | * Find the hydrostatic force | + | * Calculate the work done in stretching/compressing a spring. |
− | + | * Calculate the work done in lifting a rope/cable. | |
− | + | * Calculate the work done in pumping a liquid from one height to another. | |
+ | * Find the hydrostatic force against a submerged vertical plate. | ||
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− | |Week | + | |Week 5 |
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− | * [[Toolkit Functions| | + | * [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 --> |
− | * [[The | + | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> |
+ | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
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− | * Find the | + | * Find the center of mass of objects distributed along a line. |
− | + | * Find the center of mass of objects distributed in a plane. | |
− | + | * Locate the center of mass of a thin plate. | |
+ | * Use symmetry to help locate the centroid of a thin plate. | ||
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− | |Week 5 | + | |Week 5-6 |
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− | * [[ | + | * [[Differentiation Rules]] <!-- 1214-3.3 --> |
− | * [[Linear Approximations and Differentials| | + | * [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 --> |
− | * [[ | + | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> |
+ | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
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− | * | + | * Recognize when to use integration by parts. |
− | * | + | * Use the integration-by-parts formula to evaluate indefinite integrals. |
− | + | * Use the integration-by-parts formula to evaluate definite integrals. | |
− | + | * Use the tabular method to perform integration by parts. | |
+ | * Solve problems involving applications of integration using integration by parts. | ||
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− | |Week | + | |Week 6 |
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+ | * [[Trigonometric Functions]] <!-- 1093-2.2 --> | ||
+ | * [[Properties of the Trigonometric Functions|Trigonometric Identities]] <!-- 1093-3.4 --> | ||
+ | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
* [[Integration by Substitution]] <!-- 1224-1.5 --> | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
− | * [[ | + | * [[Integration by Parts]] <!-- 1224-3.1 --> |
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+ | * Evaluate integrals involving products and powers of sin(x) and cos(x). | ||
+ | * Evaluate integrals involving products and powers of sec(x) and tan(x). | ||
+ | * Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx). | ||
+ | * Solve problems involving applications of integration using trigonometric integrals. | ||
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− | |Week 6 | + | |Week 6-7 |
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− | * [[ | + | * [[Completing the Square]] <!-- 1073-Mod 3.2--> |
+ | * [[Trigonometric Functions]] <!-- 1093-2.2 --> | ||
+ | * [[Properties of the Trigonometric Functions|Trigonometric Identities]] <!-- 1093-3.4 --> | ||
* [[Integration by Substitution]] <!-- 1224-1.5 --> | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
+ | * [[Integration by Parts]] <!-- 1224-3.1 --> | ||
* [[Trigonometric Integrals]] <!-- 1224-3.2 --> | * [[Trigonometric Integrals]] <!-- 1224-3.2 --> | ||
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− | * | + | * Evaluate integrals involving the square root of a sum or difference of two squares. |
− | + | * Solve problems involving applications of integration using trigonometric substitution. | |
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− | |Week | + | |Week 7 |
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− | * [[Dividing Polynomials]] <!-- 1073- | + | * [[Factoring Polynomials]] <!-- 1073-Mod 0.2 --> |
+ | * [[Completing the Square]] <!-- 1073-Mod 3.2--> | ||
+ | * [[Dividing Polynomials|Long Division of Polynomials]] <!-- 1073-Mod 4.1 --> | ||
+ | * [[Systems of Linear Equations]] <!-- 1073-Mod 12.1 and 12.2 --> | ||
* [[Antiderivatives]] <!-- 1214-4.10 --> | * [[Antiderivatives]] <!-- 1214-4.10 --> | ||
− | * [[ | + | * [[Integration by Substitution]] <!-- 1224-1.5 --> |
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− | * Integrate rational | + | * Integrate a rational function whose denominator is a product of linear and quadratic polynomials. |
− | + | * Recognize distinct linear factors in a rational function. | |
+ | * Recognize repeated linear factors in a rational function. | ||
+ | * Recognize distinct irreducible quadratic factors in a rational function. | ||
+ | * Recognize repeated irreducible quadratic factors in a rational function. | ||
+ | * Solve problems involving applications of integration using partial fractions. | ||
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− | |Week | + | |Week 8 |
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− | * [[ | + | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> |
* [[Integration by Substitution]] <!-- 1224-1.5 --> | * [[Integration by Substitution]] <!-- 1224-1.5 --> | ||
* [[Integration by Parts]] <!-- 1224-3.1 --> | * [[Integration by Parts]] <!-- 1224-3.1 --> | ||
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* [[Trigonometric Integrals]] <!-- 1224-3.2 --> | * [[Trigonometric Integrals]] <!-- 1224-3.2 --> | ||
* [[Trigonometric Substitution]] <!-- 1224-3.3 --> | * [[Trigonometric Substitution]] <!-- 1224-3.3 --> | ||
− | * [[ | + | * [[Partial Fractions]] <!-- 1224-3.4 --> |
− | * [[ | + | * [[The Limit Laws]] <!-- 1214-2.3 --> |
+ | * [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1224-4.6 --> | ||
+ | * [[L’Hôpital’s Rule]] <!-- 1214-4.8 --> | ||
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− | * Recognize and | + | * Recognize improper integrals and determine their convergence or divergence. |
− | * | + | * Evaluate an integral over an infinite interval. |
− | + | * Evaluate an integral over a closed interval with an infinite discontinuity within the interval. | |
− | + | * Use the comparison theorem to determine whether an improper integral is convergent or divergent. | |
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− | * [[Limits at Infinity and Asymptotes | | + | * [[The Limit Laws| The Limit Laws and Squeeze Theorem]] <!-- 1214-2.3 --> |
− | * | + | * [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1214-4.6 --> |
+ | * [[L’Hôpital’s Rule]] <!-- 1214-4.8 --> | ||
+ | * [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] <!-- 1214-4.5 --> | ||
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− | * Find | + | * Find a formula for the general term of a sequence. |
− | * | + | * Find a recursive definition of a sequence. |
+ | * Determine the convergence or divergence of a given sequence. | ||
* Find the limit of a convergent sequence. | * Find the limit of a convergent sequence. | ||
− | * Determine whether a sequence is monotone. | + | * Determine whether a sequence is bounded and/or monotone. |
− | + | * Apply the Monotone Convergence Theorem. | |
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− | * | + | * [[Sigma notation]] <!-- DNE (recommend 1093) --> |
* [[Sequences]] <!-- 10224-5.1--> | * [[Sequences]] <!-- 10224-5.1--> | ||
+ | * [[Partial Fractions]] <!-- 1224-3.4--> | ||
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+ | * Write an infinite series using sigma notation. | ||
+ | * Find the nth partial sum of an infinite series. | ||
* Define the convergence or divergence of an infinite series. | * Define the convergence or divergence of an infinite series. | ||
− | * Find the sum of a geometric | + | * Identify a geometric series. |
+ | * Apply the Geometric Series Test. | ||
+ | * Find the sum of a convergent geometric series. | ||
+ | * Identify a telescoping series. | ||
+ | * Find the sum of a telescoping series. | ||
|- | |- | ||
− | |Week 10 | + | |Week 10-11 |
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+ | * [[The Limit Laws]] <!-- 1214-2.3 --> | ||
+ | * [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1214-4.6 --> | ||
* [[Continuity]] <!-- 1214-3.5 --> | * [[Continuity]] <!-- 1214-3.5 --> | ||
− | * [[ | + | * [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] <!-- 1214-4.5 --> |
− | * [[ | + | * [[L’Hôpital’s Rule]] <!-- 1214-4.8 --> |
− | * [[ | + | * [[Improper Integrals]] <!-- 1224-3.7 --> |
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− | * | + | * Use the Divergence Test to determine whether a series diverges. |
− | * Estimate the sum of a series | + | * Use the Integral Test to determine whether a series converges or diverges. |
− | + | * Use the p-Series Test to determine whether a series converges or diverges. | |
− | + | * Estimate the sum of a series by finding bounds on its remainder term. | |
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− | * [[ | + | * [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6--> |
− | * | + | * [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] <!-- 1214-4.5 --> |
− | * [[ | + | * [[L’Hôpital’s Rule]] <!-- 1214-4.8 --> |
+ | * [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 --> | ||
+ | * [[The Divergence and Integral Tests|The p-Series Test]] <!-- 1224-5.3 --> | ||
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− | * | + | * Use the Direct Comparison Test to determine whether a series converges or diverges. |
− | + | * Use the Limit Comparison Test to determine whether a series converges or diverges. | |
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− | |Week | + | |Week 12 |
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− | * [[ | + | * [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6--> |
− | * | + | * [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] <!-- 1214-4.5 --> |
− | * [[ | + | * [[L’Hôpital’s Rule]] <!-- 1214-4.8 --> |
+ | * [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 --> | ||
+ | * [[The Divergence and Integral Tests|The p-Series Test]] <!-- 1224-5.3 --> | ||
+ | * [[Comparison Tests]] <!-- 1224-5.4 --> | ||
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− | * | + | * Use the Alternating Series Test to determine the convergence of an alternating series. |
* Estimate the sum of an alternating series. | * Estimate the sum of an alternating series. | ||
− | * | + | * Explain the meaning of absolute convergence and conditional convergence. |
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− | * | + | * [[Factorials]] <!-- Grades 6-12 --> |
− | * [[Limits at Infinity and Asymptotes | | + | * [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6--> |
− | * [[ | + | * [[L’Hôpital’s Rule]] <!-- 1214-4.8 --> |
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− | * | + | * Use the Ratio Test to determine absolute convergence or divergence of a series. |
− | + | * Use the Root Test to determine absolute convergence or divergence of a series. | |
− | + | * Describe a strategy for testing the convergence or divergence of a series. | |
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− | |Week | + | |Week 13 |
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− | * [[ | + | * [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 --> |
− | * [[ | + | * [[The Divergence and Integral Tests]] <!-- 1224-5.3 --> |
− | * [[Series]] <!-- 1224-5. | + | * [[Comparison Tests]] <!-- 1224-5.4 --> |
+ | * [[Alternating Series]] <!-- 1224-5.5 --> | ||
* [[Ratio and Root Tests]] <!-- 1224-5.6 --> | * [[Ratio and Root Tests]] <!-- 1224-5.6 --> | ||
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− | * | + | * Identify a power series. |
− | * | + | * Determine the interval of convergence and radius of convergence of a power series. |
− | * | + | * Use a power series to represent certain functions. |
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− | |Week | + | |Week 14 |
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− | * [[Antiderivatives | + | * [[Differentiation Rules]] <!-- 1214-3.3 --> |
− | * [[The | + | * [[Antiderivatives]] <!-- 1214-4.10 --> |
+ | * [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 --> | ||
* [[Power Series and Functions]] <!-- 1224-6.1 --> | * [[Power Series and Functions]] <!-- 1224-6.1 --> | ||
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+ | * Combine power series by addition or subtraction. | ||
+ | * Multiply two power series together. | ||
* Differentiate and integrate power series term-by-term. | * Differentiate and integrate power series term-by-term. | ||
− | * | + | * Use differentiation and integration of power series to represent certain functions as power series. |
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− | |Week | + | |Week 15 |
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− | * [[The Derivative | + | * [[The Derivative as a Function|Higher-Order Derivatives]] <!-- 1214-3.2 --> |
* [[Power Series and Functions]] <!-- 1224-6.1 --> | * [[Power Series and Functions]] <!-- 1224-6.1 --> | ||
* [[Properties of Power Series]] <!-- 1224-6.2 --> | * [[Properties of Power Series]] <!-- 1224-6.2 --> | ||
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− | * Find | + | * Find a Taylor or Maclaurin series representation of a function. |
− | * Find the radius of convergence of a Taylor Series. | + | * Find the radius of convergence of a Taylor Series or Maclaurin series. |
− | * | + | * Finding a Taylor polynomial of a given order for a function. |
− | + | * Use Taylor's Theorem to estimate the remainder for a Taylor series approximation of a given function. | |
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Latest revision as of 09:39, 6 January 2024
The textbook for this course is Calculus (Volume 2) by Gilbert Strang, Edwin Herman, et al.
A comprehensive list of all undergraduate math courses at UTSA can be found here.
The Wikipedia summary of calculus and its history.
Topics List
Date | Sections | Topics | Prerequisite Skills | Student Learning Outcomes |
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Week 1 |
1.3
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Week 1 |
1.5
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Week 2 |
2.1
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Week 2 |
2.2
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Week 3 |
2.3
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Week 3 |
2.4
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Week 4 |
2.5
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Week 5 |
2.6
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Week 5-6 |
3.1
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Week 6 |
3.2
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Week 6-7 |
3.3
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Week 7 |
3.4
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Week 8 |
3.7
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Week 9 |
5.1
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Week 10 |
5.2
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Week 10-11 |
5.3
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Week 11 |
5.4
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Week 12 |
5.5
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Week 12 |
5.6
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Week 13 |
6.1
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Week 14 |
6.2
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Week 15 |
6.3
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